Problem: Solve the exponential equation for $x$. 8 5 x + 3 = ( 1 64 ) − 2 3 8\^{5x+3}=\left(\dfrac{1}{64}\right)\^{ -\frac23} $x=$
Solution: The strategy We want to rewrite one of the exponential terms in the equation so that the bases of the two terms are the same. Then, we will be able to equate the exponents and solve for $x$. [Why can we do this?] Matching the bases Let's rewrite ( 1 64 ) − 2 3 \left(\dfrac{1}{64}\right)\^{ -\frac 23} so its base is $8$. ( 1 64 ) − 2 3 = ( 8 − 2 ) − 2 3 = 8 − 2 ⋅ − 2 3 = 8 4 3 Rewrite 1 64 as 8 − 2 Since ( a n ) m = a n ⋅ m \begin{aligned} \left(\dfrac{1}{64}\right)\^{ -\frac 23} &= (8^{-2})\^{- \frac 23}&&&&\text{Rewrite } \dfrac{1}{64} \text{ as }8^{-2} \\\\ &=8\^{ -2\ \cdot\ -\frac 23} &&&&\text{Since }(a^n)^m=a^{n\cdot m}\\\\ &=8\^{ \frac 43} \end{aligned} [Can we choose another base?] Solving the equation We obtain the following equation. 8 5 x + 3 = 8 4 3 8\^{5x+3}=8\^{ \frac 43} Now we can equate the exponents and solve for $x$. $\begin{aligned}5x+3&=\dfrac 43\\\\\\ x &=-\dfrac{1}{3}\end{aligned}$ The answer The answer is $x=-\dfrac{1}{3}$. You can check this answer by substituting $\it{x=-\dfrac{1}{3}}$ in the original equation and evaluating both sides.